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Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq0.$ We suppose that $M$ is conformally flat and simply connected or moregenerally that it admits a conformal immersion into the standard 3-sphere. Let$\Sigma$ be a compact connected and orientable surface immersed in $M$ which isa stable constant mean curvature (CMC) surface or an index one minimal surface.We prove that $\Sigma$ is homeomorphic either to a sphere or to a torus.Moreover, in case $\Sigma$ is homeomorphic to a torus, then it is embedded,minimal, conformal to a flat square torus and Ric$(N)=0$ where $N$ is a unitfield normal to $\Sigma.$ The result is sharp, we can perturb the standardmetric on the 3-sphere in its conformal class to obtain metrics of nonnegativeRicci curvature admitting minimal tori which are stable as CMC surfaces.  As a consequence, in any 3-sphere of positive Ricci curvature which isconformally flat, the isoperimetric domains are topologically 3-balls. Thisproves a special case of a conjecture of A. Ros.

Stable CMC and index one minimal surfaces in conformally flat manifolds